Precise and complete control over closed quantum systems is a long-sought goal in atomic physics, molecular chemistry, condensed matter research, with fundamental implications for metrology [J. J. Bollinger, W. M. Itano, D. J. Wineland, and D. J. Heinzen, Phys. Rev. A 54, R4649, (1996); C. H. Bennett and P. W. Shor, IEEE Transactions on Information Theory, 44, 2724 (1998)] and computation [M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, UK, 2000); A. Steane, Rep. Prog. Phys. 61, 117 (1998)]. Achieving this goal will require careful compensation for errors of both random and systematic nature. While recent advances in quantum error correction [D. Gottesman, Phys. Rev. A 57, 127 (1998); A. M. Steane, Phys. Rev. Lett. 77, 793 (1996); E. Knill and R. Laflamme, Phys. Rev. A 55, 900 (1997), quant-ph/9604034] allows all such errors to be removed in principle, active error correction requires expanding the size of the quantum system, as well as feedback measurements that may be unavailable. Furthermore, in many systems, errors may be dominated by those of systematic nature, rather than by random errors, such as when the classical control apparatus is miscalibrated or suffers from inhomogeneities over the spatial extent of the target quantum system.
Such systematic errors are common in the art of nuclear magnetic resonance (NMR), where radio frequency fields (RF) applied to control nuclear spins in molecules often have pulse amplitude errors or are inhomogeneous across the sample. Of course, systematic errors can be reduced simply by calibration, but that is often impractical, especially when controlling large systems or when the required control error magnitude is smaller than what is easily measurable. Interestingly, however, systematic errors in controlling quantum systems can be compensated without specific knowledge of the magnitude of the error. This fact is lore [R. Freeman, Spin Choreography (Spektrum, Oxford, 1997)] in the art of NMR, and is achieved by using the method of composite pulses, in which a single imperfect pulse with fractional error E is replaced with a sequence of pulses, which reduces the error to O(εn).
Composite pulse sequences have been constructed to correct for a wide variety of systematic errors in NMR [R. Freeman, Spin Choreography (Spektrum, Oxford, 1997); M. Levitt and R. Freeman, J. Magn. Reson. 33, 473 (1979); R. Tycko, Phys. Rev. Lett. 51, 775 (1983)], including pulse amplitude, phase, and frequency errors. For example, in magnetic resonance imaging, composite pulse sequences can enhance frequency selectivity, and thereby increase the resolution with which spatial details can be distinguished. Generally, however, in NMR the goal is to maximize the measurable signal from a spin system that starts in a specific state. Thus, while composite sequences have been developed [M. Levitt and R. R. Ernst, J. Magn. Reson. 55, 247 (1983)] which can reduce errors to O(εn) for arbitrary n, these sequences are not general and do not apply, for example, to quantum computation, where the initial state is arbitrary and multiple operations must be cascaded in order to obtain desired multi-qubit transformations. Pulse sequences of this type have been previously patented, e.g. U.S. Pat. No. 5,153,515 (“Methods of generating pulses for selectively exciting frequencies”; Leigh et al) and U.S. Pat. No. 5,572,126 (“Reduced power selective excitation RF pulses”; Shinnar) [see also LeRoux “Method of radio-frequency excitation in an NMR experiment,” U.S. Pat. No. 4,940,940, 1990].
Only a few composite pulse sequences are known which are fully compensating [M. Levitt, Prog. in NMR Spectr. 18, 61 (1986)], meaning that they work on any initial state and can replace a single pulse without further modification of other pulses. As has been theoretically discussed [H. Cummins and J. Jones, New J. Phys. 2.6, 1 (2000); J. Jones, Phys. Rev. A 67, 012317 (2002); D. McHugh and J. Tawnley, quant-ph/0404127 (2004)] and experimentally demonstrated [S. Gulde, M. Riebe, G. P. T. Lancaster, C. Becher, J. Eschner, H. Hffner, F. Schmidt-Kaler, I. L. Chuang, and R. Blatt, Nature 421, 48 (2003)], these sequences can be valuable for quantum computation and quantum information tasks, such as precise single and multiple-qubit control using gate voltages or laser excitation. However, these pulse sequences cannot currently be effectively used to compensate for errors of both random and systematic nature. What has been needed, therefore, are composite pulses that are both fully compensating and able to correct errors arbitrarily well.